Answers and comments for Math 242 Homework 4.

#1. Final answer is 1. 

Should follow easily from doing L'Hopital's rule once.


#2. Final answer is

Pi * ( 5/2 + 4/e^2 - 1/(2e^4) )

which can also be written

Pi * ( 5e^4 + 8e^2 - 1 ) / (2e^4)

and whose approximate numerical value is

9.526

They should use WASHERS.

Top curve is y = 1, bottom curve is y = exp(-x)

"Inner radius" is 1, "outer radius" is 2 - exp(-x).

They should integrate Pi*( (2-exp(-x))^2 - 1 ) from x=0 to x=2.

The important thing is that they set up the problem correctly.

You can be forgiving if they make small algebraic mistakes
or don't show all their calculations or use Maple.


#3. Final answer is x*ln(x) - x

This follows from integration by parts. However, they've
already seen this integral, so it's fine if they just
"assert" the answer.


#4. There are several correct ways to write the final answer:

(tan x)^2 / 2 - ln|sec x| + C

(tan x)^2 / 2 + ln|cos x| + C

(sec x)^2 / 2 - ln|sec x| + C

(sec x)^2 / 2 + ln|cos x| + C

The "trick" for #4 is first to write (tan x)^3 = (tan x) (tan x)^2
and then to replace (tan x)^2 with (sec x)^2 - 1. They then get
two integrals which they should know how to do.


#5. Final answer is

(1/2) * ( sec(x)*tan(x) + ln| sec(x)+tan(x) | )

They should do integration by parts.

However, they have also seen this integral in the lectures, so
it's okay if they have the correct final answer and don't show
all the details about how they got it.


#6. Final answer is

ln(2+sqrt(3)) - ln(sqrt(2)+1)

or

ln( (2+sqrt(3))/(1+sqrt(2)) )

They don't have to show a lot of steps, because they just "know"
the antiderivative of sec x.


#7. Final answer is

(1/2)*( 2*sqrt(3) + ln(2+sqrt(3)) )

or

sqrt(3) + (1/2)*ln(2+sqrt(3))

They don't have to show a lot of steps for this one,
because they already have the antiderivative from question 5.


Note that #8 and #9 can be done on Maple and don't require
that they show all their steps.

#8. Final answer is
( -120 +120x -60x^2 +20x^3 -5x^4 +x^5 )*e^x

#9. Final answer is

-x^3*cos(x) +3x^2*sin(x) -6sin(x) +6x*cos(x)